Enhancing the Accuracy of Solving Riccati Fractional Differential Equations

Fractal and Fractional

Octavian Postavaru¹  Antonela Toma¹  Flavius Dragoi¹ 

[1] Center for Research and Training in Innovative Techniques of Applied Mathematics in Engineering, University Politehnica of Bucharest, 060042 Bucharest, Romania;

关键词: hybrid functions;    Caputo derivative;    Riemann–Liouville integral;   

DOI  :  10.3390/fractalfract6050275

来源: DOAJ

【 摘 要 】

In this paper, we solve Riccati equations by using the fractional-order hybrid function of block-pulse functions and Bernoulli polynomials (FOHBPB), obtained by replacing x with $x^{\alpha }$, with positive $\alpha$. Fractional derivatives are in the Caputo sense. With the help of incomplete beta functions, we are able to build exactly the Riemann–Liouville fractional integral operator associated with FOHBPB. This operator, together with the Newton–Cotes collocation method, allows the reduction of fractional differential equations to a system of algebraic equations, which can be solved by Newton’s iterative method. The simplicity of the method recommends it for applications in engineering and nature. The accuracy of this method is illustrated by five examples, and there are situations in which we obtain accuracy eleven orders of magnitude higher than if $\alpha =1$.

【 授权许可】

Unknown